Angles and their types

Open a door a lit­tle, and then a lot. Open a book, or watch the hands of a clock move. In each case some­thing turns, and the amount of turn­ing is an angle. In this les­son you will learn what an angle is, how to name it, how it is mea­sured in degrees, and the names of the dif­fer­ent kinds of angle.

What an angle is#

Draw two rays that start from the same point. The fig­ure they make is called an angle.

• The com­mon start­ing point is called the ver­tex of the angle.
• The two rays are called the arms of the angle.

Sup­pose the rays are $\overrightarrow{BA}$ and $\overrightarrow{BC}$, both start­ing at $B$. The angle they make is writ­ten $\angle ABC$ or $\angle CBA$. The ver­tex is always writ­ten in the mid­dle. When there is only one angle at $B$, we may sim­ply write $\angle B$.

The size of an angle depends only on how far one arm is turned from the other. It does not depend on how long the arms are drawn. Draw­ing longer arms does not make the angle big­ger.

Mea­sur­ing angles in degrees#

Angles are mea­sured in degrees, writ­ten with a small cir­cle: $1^\circ$ is one degree.

Think of one arm turn­ing all the way round until it comes back to where it started. That is one com­plete turn, and a com­plete turn is $360^\circ$.

A pro­trac­tor is used to mea­sure angles. To mea­sure $\angle ABC$:

1. Put the cen­tre point of the pro­trac­tor exactly on the ver­tex $B$.
2. Line up the base line of the pro­trac­tor with the arm $BC$.
3. Start from the zero on that arm and read along the same scale to where the arm $BA$ crosses it.

A pro­trac­tor has two scales, one going each way. Always use the scale that starts at zero on the arm you lined up.

Kinds of angles#

A right angle is the cor­ner of a page or of a square tile. It is often marked with a small square drawn at the ver­tex. Two lines that meet at a right angle are called per­pen­dic­u­lar. We write $l \perp m$.

Exam­ple 1#

Name the kind of each angle: $35^\circ$, $90^\circ$, $145^\circ$, $180^\circ$, $210^\circ$ and $360^\circ$.

• $35^\circ$ is between $0^\circ$ and $90^\circ$, so it is acute.
• $90^\circ$ is a right angle.
• $145^\circ$ is between $90^\circ$ and $180^\circ$, so it is obtuse.
• $180^\circ$ is a straight angle.
• $210^\circ$ is between $180^\circ$ and $360^\circ$, so it is reflex.
• $360^\circ$ is a com­plete angle.

The reflex angle on the other side#

Two arms from one ver­tex really make two angles: the smaller one between them, and the larger one going the long way round. Together they make a com­plete turn. So

$$\text{reflex angle} = 360^\circ - \text{the smaller angle}$$

Exam­ple 2#

The angle between two arms is $130^\circ$. Find the reflex angle between them.

$$360^\circ - 130^\circ = 230^\circ$$

When we just say "the angle between two arms", we mean the smaller one, unless the ques­tion says reflex.

Angles on a clock face#

A clock face is a cir­cle of $360^\circ$ split into 12 equal hour spaces. So each hour space is

$$360^\circ \div 12 = 30^\circ$$

At an exact hour, the minute hand points at 12. Count the hour spaces between the two hands and mul­ti­ply by $30^\circ$.

Exam­ple 3#

Find the angle between the hands of a clock at 2 o'clock, at 3 o'clock, at 5 o'clock and at 6 o'clock.

• At 2 o'clock the hands are 2 spaces apart: $2 \times 30^\circ = 60^\circ$, an acute angle.
• At 3 o'clock: $3 \times 30^\circ = 90^\circ$, a right angle.
• At 5 o'clock: $5 \times 30^\circ = 150^\circ$, an obtuse angle.
• At 6 o'clock: $6 \times 30^\circ = 180^\circ$, a straight angle.

Exam­ple 4#

Find the angle between the hands at 8 o'clock.

Count­ing for­ward from 12 to 8 gives 8 spaces, that is $240^\circ$. But going the short way, from 8 on to 12, there are only 4 spaces. The angle between the hands is $4 \times 30^\circ = 120^\circ$. The reflex angle is $240^\circ$.

Turn­ing and direc­tions#

If you face north and turn clock­wise to face east, you turn through a right angle, $90^\circ$. North to south is $180^\circ$. Turn­ing from north all the way round back to north is $360^\circ$.

Exam­ple 5#

Ravi faces east and turns clock­wise through $180^\circ$. Which way does he face now? What if he turns another $90^\circ$ clock­wise?

Clock­wise from east the order is east, south, west, north, with $90^\circ$ between neigh­bours. A turn of $180^\circ$ is two steps, so he faces west. Another $90^\circ$ clock­wise brings him to north.

Prac­tice#

1. Name the ver­tex and the arms of $\angle PQR$.
2. Name the kind of each angle: $89^\circ$, $91^\circ$, $179^\circ$, $181^\circ$.
3. How many degrees are there in a quar­ter turn? In three quar­ters of a turn?
4. The angle between two arms is $75^\circ$. Find the reflex angle between them.
5. A reflex angle is $300^\circ$. What is the smaller angle between the same two arms?
6. Find the angle between the hands of a clock at 4 o'clock.
7. Find the angle between the hands of a clock at 10 o'clock.
8. Sita faces north and turns clock­wise through $270^\circ$. Which way does she face?
9. Does a longer pair of arms make an angle big­ger? Explain.

Answers#